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{{Algebraic structures|module}}

In [[mathematics]], a '''module''' is a generalization of the notion of [[vector space]] in which the [[Field (mathematics)|field]] of [[scalar (mathematics)|scalars]] is replaced by a (not necessarily [[Commutative ring|commutative]]) [[Ring (mathematics)|ring]]. The concept of a ''module'' also generalizes the notion of an [[abelian group]], since the abelian groups are exactly the modules over the ring of [[integer]]s.<ref>Hungerford (1974) ''Algebra'', Springer, p 169: "Modules over a ring are a generalization of abelian groups (which are modules over Z)."</ref>

Like a vector space, a module is an additive abelian group, and scalar multiplication is [[Distributive property|distributive]] over the operations of addition between elements of the ring or module and is [[Semigroup action|compatible]] with the ring multiplication.

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In a vector space, the set of [[scalar (mathematics)|scalars]] is a [[field (mathematics)|field]] and acts on the vectors by scalar multiplication, subject to certain axioms such as the [[distributive law]]. In a module, the scalars need only be a [[ring (mathematics)|ring]], so the module concept represents a significant generalization. In commutative algebra, both [[ideal (ring theory)|ideals]] and [[quotient ring]]s are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.<!-- (semi)perfect rings for instance have a litany of "Foo is true for all left ideals iff foo is true for all finitely generated left ideals iff foo is true for all cyclic modules iff foo is true for all modules" -->

Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "[[well-behaved]]" ring, such as a [[principal ideal ___domain]]. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a [[basis (linear algebra)|basis]], and, even for those that do ([[free module]]s), the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique [[Free_module#Definition|rank]]) if the underlying ring does not satisfy the [[invariant basis number]] condition, unlike vector spaces, which always have a (possibly infinite) basis whose [[cardinality]] is then unique. (These last two assertions require the [[axiom of choice]] in general, but not in the case of [[finite-dimensional]] vector spaces, or certain well-behaved infinite-dimensional vector spaces such as [[Lp space|L^''p'' space]]s.)

=== Formal definition ===

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The operation · is called ''scalar multiplication''. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in ''R''. One may write _''R''''M'' to emphasize that ''M'' is a left ''R''-module. A '''right ''R''-module''' ''M''_''R'' is defined similarly in terms of an operation {{nowrap|· : ''M'' × ''R'' → ''M''}}.

The qualificative of left- or right-module does not depend on whether the scalars are written on the left or on the right, but on the property 3: if, in the above definition, the property 3 is replaced by

:<math> ( r s ) \cdot x = s \cdot ( r \cdot x ), </math>

one gets a right-module, even if the scalars are written on the left. However, writing the scalars on the left for left-modules and on the right for right modules makes the manipulation of property 3 much easier.

Authors who do not require rings to be [[unital algebra|unital]] omit condition 4 in the definition above; they would call the structures defined above "unital left ''R''-modules". In this article, consistent with the [[glossary of ring theory]], all rings and modules are assumed to be unital.<ref name="DummitFoote">{{cite book | title=Abstract Algebra | publisher=John Wiley & Sons, Inc. |author1=Dummit, David S. |author2=Foote, Richard M. |name-list-style=amp | year=2004 | ___location=Hoboken, NJ | isbn=978-0-471-43334-7}}</ref>

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An (''R'',''S'')-[[bimodule]] is an abelian group together with both a left scalar multiplication · by elements of ''R'' and a right scalar multiplication ∗ by elements of ''S'', making it simultaneously a left ''R''-module and a right ''S''-module, satisfying the additional condition {{nowrap|1=(''r'' · ''x'') ∗ ''s'' = ''r'' ⋅ (''x'' ∗ ''s'')}} for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''.

If ''R'' is [[commutative ring|commutative]], then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules. Most often the scalars are written on the left in this case.

== Examples ==

*If ''K'' is a [[field (mathematics)|field]], then ''K''-modules are called ''K''-[[vector space]]s (vector spaces over ''K'') and ''K''-modules are identical.

*If ''K'' is a field, and ''K''[''x''] a univariate [[polynomial ring]], then a [[Polynomial ring#Modules|''K''[''x'']-module]] ''M'' is a ''K''-module with an additional action of ''x'' on ''M'' by a group homomorphism that commutes with the action of ''K'' on ''M''. In other words, a ''K''[''x'']-module is a ''K''-vector space ''M'' combined with a [[linear map]] from ''M'' to ''M''. Applying the [[structure theorem for finitely generated modules over a principal ideal ___domain]] to this example shows the existence of the [[Rational canonical form|rational]] and [[Jordan normal form|Jordan canonical]] forms.

*The concept of a '''Z'''-module agrees with the notion of an abelian group. That is, every [[abelian group]] is a module over the ring of [[integer]]s '''Z''' in a unique way. For {{nowrap|''n'' > 0}}, let {{nowrap|1=''n'' ⋅ ''x'' = ''x'' + ''x'' + ... + ''x''}} (''n'' summands), {{nowrap|1=0 ⋅ ''x'' = 0}}, and {{nowrap|1=(−''n'') ⋅ ''x'' = −(''n'' ⋅ ''x'')}}. Such a module need not have a [[basis (linear algebra)|basis]]—groups containing [[torsion element]]s do not. (For example, in the group of integers [[modular arithmetic|modulo]] 3, one cannot find even one element that satisfies the definition of a [[linearly independent]] set, since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a [[finite field]] is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)

*The [[decimal fractions]] (including negative ones) form a module over the integers. Only [[singleton (mathematics)|singletons]] are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no [[rank of a free module|rank]], in the usual sense of linear algebra. However this module has a [[torsion-free rank]] equal to 1.

*If ''R'' is any ring and ''n'' a [[natural number]], then the [[cartesian product]] ''R''^''n'' is both a left and right ''R''-module over ''R'' if we use the component-wise operations. Hence when {{nowrap|1=''n'' = 1}}, ''R'' is an ''R''-module, where the scalar multiplication is just ring multiplication. The case {{nowrap|1=''n'' = 0}} yields the trivial ''R''-module {0} consisting only of its identity element. Modules of this type are called [[free module|free]] and if ''R'' has [[invariant basis number]] (e.g. any commutative ring or field) the number ''n'' is then the rank of the free module.

*If M_''n''(''R'') is the ring of {{nowrap|''n''&thinsp;×&thinsp;''n''}} [[matrix (mathematics)|matrices]] over a ring ''R'', ''M'' is an M_''n''(''R'')-module, and ''e''_''i'' is the {{nowrap|''n'' × ''n''}} matrix with 1 in the {{nowrap|(''i'', ''i'')}}-entry (and zeros elsewhere), then ''e''_''i''''M'' is an ''R''-module, since {{nowrap|1=''re''_''i''''m'' = ''e''_''i''''rm'' ∈ ''e''_''i''''M''}}. So ''M'' breaks up as the [[direct sum]] of ''R''-modules, {{nowrap|1=''M'' = ''e''₁''M'' ⊕ ... ⊕ ''e''_''n''''M''}}. Conversely, given an ''R''-module ''M''₀, then ''M''₀^⊕''n'' is an M_''n''(''R'')-module. In fact, the [[category of modules|category of ''R''-modules]] and the [[category (mathematics)|category]] of M_''n''(''R'')-modules are [[equivalence of categories|equivalent]]. The special case is that the module ''M'' is just ''R'' as a module over itself, then ''R''^''n'' is an M_''n''(''R'')-module.

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Suppose ''M'' is a left ''R''-module and ''N'' is a [[subgroup]] of ''M''. Then ''N'' is a '''submodule''' (or more explicitly an ''R''-submodule) if for any ''n'' in ''N'' and any ''r'' in ''R'', the product {{nowrap|''r'' ⋅ ''n''}} (or {{nowrap|''n'' ⋅ ''r''}} for a right ''R''-module) is in ''N''.

If ''X'' is any [[subset]] of an ''R''-module ''M'', then the submodule spanned by ''X'' is defined to be <math display="inline">\langle X \rangle = \,\bigcap_{N\supseteq X} N</math> where ''N'' runs over the submodules of ''M'' that contain ''X'', or explicitly <math display="inline">\leftbigl\{\!\sum_{i=1}^k r_ix_i \midmathrel{\big|} r_i \in R,\, x_i \in X\rightbigr\}</math>, which is important in the definition of [[tensor product of modules|tensor products of modules]].<ref>{{Cite web|url=http://people.maths.ox.ac.uk/mcgerty/Algebra%20II.pdf|title=ALGEBRA II: RINGS AND MODULES|last=Mcgerty|first=Kevin|date=2016}}</ref>

The set of submodules of a given module ''M'', together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a [[Lattice (order)|lattice]] that satisfies the '''[[modular lattice|modular law]]''':

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If ''M'' is a left ''R''-module, then the ''action'' of an element ''r'' in ''R'' is defined to be the map {{nowrap|''M'' → ''M''}} that sends each ''x'' to ''rx'' (or ''xr'' in the case of a right module), and is necessarily a [[group homomorphism|group endomorphism]] of the abelian group {{nowrap|(''M'', +)}}. The set of all group endomorphisms of ''M'' is denoted End_'''Z'''(''M'') and forms a ring under addition and [[function composition|composition]], and sending a ring element ''r'' of ''R'' to its action actually defines a [[ring homomorphism]] from ''R'' to End_'''Z'''(''M'').

Such a ring homomorphism {{nowrap|''R'' → End_'''Z'''(''M'')}} is called a ''representation'' of the abelian group ''RM'' over the abelian groupring ''MR''; an alternative and equivalent way of defining left ''R''-modules is to say that a left ''R''-module is an abelian group ''M'' together with a representation of ''RM'' over it''R''. Such a representation {{nowrap|''R'' → End_'''Z'''(''M'')}} may also be called a ''ring action'' of ''R'' on ''M''.

A representation is called ''faithful'' if and only if the map {{nowrap|''R'' → End_'''Z'''(''M'')}} is [[injective]]. In terms of modules, this means that if ''r'' is an element of ''R'' such that {{nowrap|1=''rx'' = 0}} for all ''x'' in ''M'', then {{nowrap|1=''r'' = 0}}. Every abelian group is a faithful module over the [[integer]]s or over somethe [[Modular arithmetic|ring of integers modulo ''n'']], '''Z'''/''n'''''Z''', for some ''n''.

=== Generalizations ===